Structural rigidity of generalised Volterra operators on Hp

We show that the non-compact generalised analytic Volterra operators Tg, where g∈BMOA, have the following structural rigidity property on the Hardy spaces Hp for 1≤p<∞ and p≠2: if Tg is bounded below on an infinite-dimensional subspace M⊂Hp, then M contains a subspace linearly isomorphic to ℓp. This implies in particular that any Volterra operator Tg:Hp→Hp is ℓ2-singular for p≠2.